We first consider the case of trivial coupling phase \(\theta =0\)  and plot in Figs. REF (a)-REF (c) the dynamical evolutions of \(P_{b,1}\)  and \(P_{a,2}\)  with \(\eta =0.56\) , \(\eta =0.574\) , and \(\eta =0.588\) , respectively. It shows that the waveguide induced phase factor \(\phi \)  cannot lead to nonreciprocity since it does not break the time-reversal symmetry of the Hamiltonian. Note that the populations decay much slower when \(\phi \)  is an integer multiple of \(\pi \) . This is reminiscent of the Fabry-Pérot bound states in the continuum (BICs) in the Markovian limit, which shows that one of the eigenstates becomes lossless if \(\phi =m\pi \)  (\(m\)  is an arbitrary integer) [1]}, [2]}. For general initial states considered here, one can find from Eq. (REF ) that the waveguide induced indirect coupling is purely dissipative (i.e., \(i\gamma e^{i\phi }/2\)  is purely imaginary) in the case of \(\phi =m\pi \) , which serves as an effective gain and thus suppresses the decay of the emitters [3]}. Moreover, the populations decay in an oscillating form for \(\phi \ne m\pi \)  because the excitation bounces between the emitters back and forth.
